Technically, one of the most effective tools turned out to be matrix models, working out these simple examples it was possible to develop a notion of check-operators. These operators act on the moduli space, and it turned out to be possible to mimic the action of various operators of the concrete theory by an action of check-operators [1]}, [2]}. This framework allowed one later to realize many essential structures, from the topological recursion[3]}, [4]}, [5]}, [6]} to dualities [7]}, from the wall-crossing formulas [8]} to knot theory [8]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}. Moreover, it turns out that this approach provides a simple description of some standard objects like modular kernels [7]} and of some standard procedures like quantization of Seiberg–Witten integrabilities [17]}, both the standard [18]}, [19]}, [20]} and Whitham [21]}, [22]}, [17]} ones. As predicted [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, nowadays they proved important far beyond the matrix model context, where they were originally discovered. The same will definitely happen to the new insights of general value from matrix model theory, which we mostly restrict
to in the present review. While 25 years ago one mostly dealt with integrability, Virasoro constraints (or loop equations, or Ward identities) genus expansions and spectral curves, today the top issues are check-operators, their action on spaces of solutions to the loop equations, the quantum spectral curves and quantized Whitham flows.
